Hopfions and Hopf Invariant: Twisted Geometry of Space Curves

Topological invariants such as winding numbers and linking numbers appear in diverse physical systems described by a three-component unit vector field on 1D, 2D and 3D manifolds. We map the vector field to tangents of appropriate space curves that depict the manifold concerned. We then invoke the concept of parallel transport of the Frenet-Serret frame of space curves and the associated anholonomy. Our analysis shows that various topological invariants emerge in a natural fashion, as integrals of distinct intrinsic geometric quantities described by torsions (signifying nonplanarity) of the above space curves, as well as their intrinsic twists. We find that the presence of intrinsic twists plays a crucial role in the nontrivial topological invariants such as Hopf invariants (H) that arise in 3D manifolds. We illustrate this by considering exact hopfion solutions of an inhomogeneous, anisotropic 3D Heisenberg ferromagnet and show that a certain intrinsic twist is indeed necessary to yield a nontrivial H.